Dynamics of Machines - Part II - IFS.pdf

% Lumped Masses
m1 = 1.95 % [kg] lowest mass
m2 = 1.72 % [kg] above lowest mass
m3 = 1.95 % [kg] below highest mass
m4 = 3.04 % [kg] highest mass with motor
% Beam Properties
E=2.0e11 % [N/m^2] elasticity modulus
b=0.0291 % [m] width
h=0.0011 % [m] thickness
I=(b*h^3)/12 % [m^4] area moment of inertia
% Position of the Platforms
L1=0.122 % [m] Length to Platform 1
L2=0.123 % [m] Length from Platform 1 to Platform 2
L3=0.149 % [m] Length from Platform 2 to Platform 3
L4=0.127 % [m] Length from Platform 3 to Platform 4
• Write the mathematical model (equations of motion) based on Newton’s laws, achieving
the mass M, stiffness K and damping D matrices.
• How is the structure of the mass matrix M? How to properly calculate the mass
coefficients?
• How is the structure of the stiffness matrix K? Find two different ways of obtaining
the stiffness coefficients.
• How is the structure of the damping matrix D? Find three different ways of obtaining
the damping coefficients, using proportional damping,
D1 = α · M.
D2 = β · K.
D3 = α · M + β · K.
Remember that when you do not know how to exactly model and achieve the damping
coefficients of the matrix D, assumptions have to be made. A realistic approximation
of the structural damping factor ξ is 0.005, according to the experimental results
obtained in equation (46). Try to adjust the proportionality factors α and β so that
the damping factor ξ1 related to the first mode shape of the structure is 0.005. Feel
free to do your own assumptions regarding damping, if you want!
2. NUMERICAL IMPLEMENTATION – Write a program in Matlab dof4-integration.m, or
use the program of your preference. Use as reference the dof2-integration.m, if you want.
Calculate the natural frequencies ωi (i = 1, ...,4) of the mechanical model and the damping
ratios ξi (i = 1, ...,4) related to the 4 modes shapes, considering the three different damping
matrices D1, D2 and D3.
%State Matrices A and B
A= [ M D ;
zeros(size(M)) M ] ;
B= [ zeros(size(M)) K ;
-M zeros(size(M))];
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